29.7.27 problem 202

Internal problem ID [4802]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 7
Problem number : 202
Date solved : Monday, January 27, 2025 at 09:39:58 AM
CAS classification : [_separable]

\begin{align*} x y^{\prime }&=\left (-2 x^{2}+1\right ) \cot \left (y\right )^{2} \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 22

dsolve(x*diff(y(x),x) = (-2*x^2+1)*cot(y(x))^2,y(x), singsol=all)
 

\[ x^{2}-y \left (x \right )-\ln \left (x \right )+\frac {\pi }{2}+c_{1} +\tan \left (y \left (x \right )\right ) = 0 \]

✓ Solution by Mathematica

Time used: 0.534 (sec). Leaf size: 55

DSolve[x D[y[x],x]==(1-2 x^2)Cot[y[x]]^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {1}{2} (\tan (\text {$\#$1})-\arctan (\tan (\text {$\#$1})))\&\right ]\left [-\frac {x^2}{2}+\frac {\log (x)}{2}+c_1\right ] \\ y(x)\to -\frac {\pi }{2} \\ y(x)\to \frac {\pi }{2} \\ \end{align*}